Here \(A\)  is a nonnegative \(d \times n\)  matrix of rank \(d\)  with no zero column,
\(c \in \mathbb {R}^n\)  is a row vector, and
\(b \in \mathbb {R}^d\)  is a column vector.
This program is feasible if and only if \(b\)  lies in
\({\rm pos}(A)\) , which is the convex polyhedral cone spanned by the columns of \(A\) .
If \(c\)  is fixed and generic, and \(b\)  ranges over \({\rm pos}(A)\) , then the set of optimal bases of (REF )
defines a regular triangulation of the cone \({\rm pos}(A)\) .
This classical result due to Walkup and Wets is explained geometrically in [1]}.
The triangulation is replaced by a continuous shape under a regularization
\({\rm Minimize}\, \,\,c \cdot x \,+\, \epsilon \sum _{i=1}^n H(x_i) \,\,\,\,\hbox{subject to} \,\,\,A x = b\,\,{\rm and} \,\, x \ge 0 .\) 
